Integrand size = 21, antiderivative size = 291 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=-8 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+2 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \]
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\[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.07 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=-8 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}+\frac {\log (1+x) \left (\log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right )-\log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )\right )}{\sqrt {2}}+\sqrt {2} \left (-\log \left (-1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )+\log \left (1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )+\log \left (-1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-\log \left (1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-\operatorname {PolyLog}\left (2,-\left (\left (-1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )+\operatorname {PolyLog}\left (2,\left (-1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )-\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \,\operatorname {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )\) | \(172\) |
default | \(8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \,\operatorname {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )\) | \(172\) |
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Exception generated. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \]
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none
Time = 0.29 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.26 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\frac {1}{2} \, {\left (\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}\right ) - \frac {4}{\sqrt {\sqrt {x + 1} + 1}}\right )} \log \left (x + 1\right ) + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} + 1\right ) + 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} - 1\right ) \]
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\[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\log \left (x + 1\right )}{x \sqrt {\sqrt {x + 1} + 1}} \,d x } \]
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Timed out. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {\ln \left (x+1\right )}{x\,\sqrt {\sqrt {x+1}+1}} \,d x \]
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